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User blog:Edwin Shade/In The Pursuit Of Organization
At the moment I have over 200 papers in my room, almost all of which contain a scrap of math on them. To explain how my stack of papers grew to this point, I should mention that I like to keep everything as organized as possible, yet lately I have not been allocating enough time for this, and so naturally things fell into a slight state of disrepair. I could not stand this for long though, because I am of the sort who cannot think as efficiently when my surroundings are cluttered as when things are well-organized, so I decided to compile all my dicoveries into this one blog post, which from now on will be the "Hub" of those mathematics which have lain for too long on the floor of my room. There are some things in here which may prove interesting, such as the pattern I believe I discovered within the prime numbers, and a way to encode the information for almost any conceivable origami creation in just one symbol. For the most part though, this blog post will be incomplete for now, and will only reach it's full completion after transcribing everything I felt was worth sharing in those papers. Feel free to comment below and ask questions if you want to. The Leading Digit of \(a^b\) \(L(a^{b})=rounddown\Big[{10^{rounddown{b(log(a))} - {b(log(a))}}}\Big]\) Hyper Subscription Hyper Subscription is denoted by two adjacent underscores, _ _, and is to denote successive multiple subscriptions. G_{G_{G_{G_{64}}}} can be rewritten as (G_ _4)_64. For a large row of subscripts this is more practical, and also allows us to define ordinals such as \(epsilon_0\) simply as (\epsilon_ _{\omega})_0. Extension of the Differential and Integral Calculus For Hyperoperators From what I have gathered, Calculus is the study of curves, slopes, and the area subtended by these curves. It is a wonder to me why no one seems to have extended Calculus to Googological functions, so I have attempted to extend traditional calculus. First however, I will present a derivation of the area underneath a standard parabola between any two intervals which I discovered about two years ago. First we will consider the area underneath the graph \(f(x)=x^2\) in the interval of \(0,4\), on the x-axis that is. The area between these two intervals and the x-axis is equal to the average sum of all points on the curve multiplied by the length of the interval, in this case, 4. I proved this by considering the case of a line running through the plane at an angle. The average height of the points on this line, (or the horizontal line that bisects it in two), multiplied by the interval the line runs over is equal to the area subtended by the line. Therefore, we can consider a curve to consist of an infinite number of extremely tiny straight lines, each one subtending a region that is like a rectangle, but also a line at the same time. To find the average height of a point on the graph of \(f(x)=x^2\) between the x-values 0 and 4 inclusive, I first considered a single line segment extending from \(f(0)\) to \(f(4)\), or 0 and 16. Such a line has an average height equivalent to \(\frac{0^2 +4^2}{2}=8\). By multiplying this by the interval 4 we get an area of 32, which might suffice for practical purposes, but is certainly not very accurate ! We have to find the average of all the point on the curve. So we again split it up into a line segment, but this time two, one extending from \(f(0)\) to \(f(2)\) and the other from \(f(2)\) to \(f(4)\). There are three unique points to consider so I calculate \(\frac{0^2 +2^2 +4^2}{3}\), giving me the average height of this group of line segments, by which I can calculate the area subtended to be \(26\frac{2}{3}\). By treating the curve as a succession of line segments that increase exponentially in number, it is possible to create a general formula for the area of this graph. Paradoxicals Paradoxicals are a new type of number I invented meant to be the solution to problems which entail division by 0. Naturally, as \(i\) is the solution to \({x^2}+1=0\), \(\rho\) is to be the solution to \(x-{\frac{1}{0}}=0\), which is currently undefined. Some Properties of \(\rho\) More information will be typed here as I discover more of the properties of paradoxicals. Proof \(a\uparrow\uparrow{\infty}\) Converges for \(a\leq {e^{\frac{1}{e}}}\) First, let \(a^{a^{a^{.^{.^{.}}}}}=x\). From this, it follows that \(x=a^x\). Now take the x-th root of both sides. \(x^{\frac{1}{x}}=a\) Calculate the derivative of a. \(a'=\frac{d}{dx}x^{\frac{1}{x}}\) \(=\frac{d}{dx}e^{\frac{1}{x}(ln(x))}\) \(={x^{\frac{1}{x}-2}}(ln(x)-1)\) Now set the derivative equal to 0 so the maxima of \(a=x^{\frac{1}{x}}\) can be found. \(x^{\frac{1}{x}-2}(ln(x)-1)\) \(ln(x)-1=0\), the other part of the equation may be crossed out because if we find an x such that \(ln(x)-1=0\) the value of \(x^{\frac{1}{x}-2}\) will be irrelevant. \(ln(x)=1\) \(x=e\) Now insert \(e\) in the expression \(x^{\frac{1}{x}}\). \(a=e^{\frac{1}{e}}\), therefore the maxima of x is \(e^{\frac{1}{e}}\). Hence, the finite maxima of \(a\uparrow\uparrow\infty\) is \(e\). Q.E.D. Transcendental Origami Transcendental origami is a form of origami I have created which entails either an infinite amount of paper or an infinite number of folds. Clearly, the structures possible within transcendental origami go beyond anything possible in the real world, hence the name. Structures That Require An Infinite Number Of Folds Origami structures whose fold numbers can be mapped to the set of natural numbers have a \(\aleph_0\) number of folds. There are however structures with more folds than these, such as the folding of a curve, which as it must consist of an infinitude of creases tangential to every point on the curve and hence have a number of folds equal to the number of points of the curve, the number of folds is equivalents to the number of real numbers, or \(\beth_1\). Structures That Require An Infinite Number Of Modules be continued ! A Pattern In The Prime Numbers Arrange the prime numbers in ascending succession, beginning with 2. 2 3 5 7 11 13 17 19 23 29 31 . . . Now take the absolute differences of successive primes, and take the absolute differences of these differences ad infinitum. 2 3 5 7 11 13 17 19 23 29 31 . . . 1 2 2 4 2 4 2 4 6 2 1 0 2 2 2 2 2 2 4 1 2 0 0 0 0 0 2 1 2 0 0 0 0 2 1 2 0 0 0 2 1 2 0 0 2 1 2 0 2 1 2 2 1 0 1 Observe the diagonal of 1's. I hypothesize this to be unceasing in it's regularity. To continue, let us list out the squares of successive primes, and take the absolute differences all the way down as we did before ad infinitum. 4 9 25 49 121 169 289 361 529 841 961 1369 1681 1849 2209 2809. . . 5 16 24 72 48 120 72 168 312 120 408 312 168 160 600 11 8 48 24 72 48 96 144 192 288 96 144 8 440 3 40 24 48 24 48 48 48 96 192 48 136 432 37 16 24 24 24 0 0 48 96 144 88 296 21 8 0 0 24 0 48 48 48 56 208 13 8 0 24 24 48 0 0 8 152 5 8 24 0 24 48 0 8 144 3 16 24 24 24 48 8 136 13 8 0 0 24 40 128 5 8 0 24 16 88 3 8 24 8 72 5 16 16 64 11 0 48 11 48 37 As LittlePeng9 has noted in the comments section, this does not appear to yeild regular results. The conclusion in this is that the primes appear only to have this regular pattern in taking the additive differences of themselves, rather than of their squares, cubes, or any other expression. Encoding (Almost) Any Origami Creation In One Symbol This is a scheme to encode any well-defined origami creation in just one symbol, albeit a large number of those symbols. First, here is the notation with which I will use to notate folds. You may notice it is based off of set-builder notation, and that is intentional. The Notation \(\) means that an aspect A is defined as \(\phi\). Pointed brackets are to be used for definitions. \(\{A|\phi\}\) means to fold an aspect A of the structure such that the condition \(\phi\) is satisfied, then unfold. \(\phi\) means to fold an aspect A of the structure according to the condition \(\phi\) and keep it folded. \(\mapsto\) means touching, or meeting. For instance, if I had the statement \(\{c1|p3\mapsto p4\}\) it would read as "fold and unfold a crease, crease-one, such that point-three meets point-4". \(pn\) means the n-th point. Points are labeled with decimal numbers after them. \(ln\) means the n-th line. Lines are labeled with decimal numbers after them. \(sn\) means the n-th surface. Surfaces are labeled with decimal numbers after them. \(cn\) means the n-th crease. Creases are labeled with decimal numbers after them. \(fn\) means the n-th fold. Note that folds are different from creases in that although every fold must contain at least one crease, a fold may contain multiple creases, (like a petal fold), whereas a crease is just one fold. Folds are labeled with decimal numbers after them. \(A\cap B\) means the intersection of aspects A and B. \(A\cup B\) means the union of aspects A and B, or in a more specific case, the area on the paper bounded by the aspects A and B. If A and B are points then \(A\cup B\) describes a line. \(|\) means such that and has the same definition as it does in set-builder notation. \((F)\) means to flip over the origami structure as a whole. \(A|\phi\land\psi\) means to perform an action A such that the conditions \(\phi\) and \(\psi\) are both satisfied. The four points of a square sheet of paper from the top left clockwise to the bottom left are to be assigned the names A, B, C, and D, which will be the four main points from which all valid origami structures are derived from. Examples \(\{c1|(A\mapsto C)\land(B\mapsto D)\}\) indicates a folding of the paper from the top down horizontally, then unfolding - as shown in the picture. What Has Been Cluttering My Room I own an awful habit of writing down undone drawths, then stowing the paper away somewhere, and then writing down the same exact thing on a separate sheet of paper again because I have not the time to swiftly sift through my papers for the original paper. So as to remedy this situation once and for all I have decided to throw away all loose papers in my room - but with a caveat. I will be translating all these loose papers into photos, so that they might be contained within one easily accessible place, albeit a less tangible one than in physical print. Thus the following plethora of pictures should give some indication overall of the sporadic and non-commitment attitude I have at times towards mathematics,but which I hope to change by this step in organization. Slowly yet surely I have whittled away the days of maze-esque pesky papers, whose passive presence below show the extent of the mess which I had gotten in. Category:Blog posts